Properties

Degree 2
Conductor $ 3 \cdot 17 \cdot 157 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2.18·2-s − 3-s + 2.78·4-s + 0.653·5-s − 2.18·6-s + 0.500·7-s + 1.71·8-s + 9-s + 1.42·10-s − 2.54·11-s − 2.78·12-s + 4.91·13-s + 1.09·14-s − 0.653·15-s − 1.82·16-s − 17-s + 2.18·18-s − 5.72·19-s + 1.81·20-s − 0.500·21-s − 5.57·22-s + 1.05·23-s − 1.71·24-s − 4.57·25-s + 10.7·26-s − 27-s + 1.39·28-s + ⋯
L(s)  = 1  + 1.54·2-s − 0.577·3-s + 1.39·4-s + 0.292·5-s − 0.892·6-s + 0.189·7-s + 0.605·8-s + 0.333·9-s + 0.452·10-s − 0.768·11-s − 0.803·12-s + 1.36·13-s + 0.292·14-s − 0.168·15-s − 0.455·16-s − 0.242·17-s + 0.515·18-s − 1.31·19-s + 0.406·20-s − 0.109·21-s − 1.18·22-s + 0.219·23-s − 0.349·24-s − 0.914·25-s + 2.10·26-s − 0.192·27-s + 0.263·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8007\)    =    \(3 \cdot 17 \cdot 157\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8007} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8007,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;17,\;157\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;17,\;157\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
17 \( 1 + T \)
157 \( 1 + T \)
good2 \( 1 - 2.18T + 2T^{2} \)
5 \( 1 - 0.653T + 5T^{2} \)
7 \( 1 - 0.500T + 7T^{2} \)
11 \( 1 + 2.54T + 11T^{2} \)
13 \( 1 - 4.91T + 13T^{2} \)
19 \( 1 + 5.72T + 19T^{2} \)
23 \( 1 - 1.05T + 23T^{2} \)
29 \( 1 + 6.02T + 29T^{2} \)
31 \( 1 - 5.10T + 31T^{2} \)
37 \( 1 + 4.78T + 37T^{2} \)
41 \( 1 + 5.46T + 41T^{2} \)
43 \( 1 + 4.78T + 43T^{2} \)
47 \( 1 + 11.0T + 47T^{2} \)
53 \( 1 - 6.18T + 53T^{2} \)
59 \( 1 - 3.15T + 59T^{2} \)
61 \( 1 + 1.40T + 61T^{2} \)
67 \( 1 - 12.7T + 67T^{2} \)
71 \( 1 - 12.0T + 71T^{2} \)
73 \( 1 + 12.6T + 73T^{2} \)
79 \( 1 - 5.61T + 79T^{2} \)
83 \( 1 + 7.83T + 83T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 + 9.67T + 97T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.09629478122703357833067216552, −6.37620891574341891004843499542, −6.08954473849164262671086233555, −5.23070951109254695056702019653, −4.85992074027149860979537978255, −3.91727926976971538225073193326, −3.46449526564174255908760487028, −2.35043846587133465401191869713, −1.63376093141875753539341523652, 0, 1.63376093141875753539341523652, 2.35043846587133465401191869713, 3.46449526564174255908760487028, 3.91727926976971538225073193326, 4.85992074027149860979537978255, 5.23070951109254695056702019653, 6.08954473849164262671086233555, 6.37620891574341891004843499542, 7.09629478122703357833067216552

Graph of the $Z$-function along the critical line